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A forward-backward SDE from the 2D nonlinear stochastic heat equation

Publication ,  Journal Article
Dunlap, A; Gu, Y
Published in: Annals of Probability
May 1, 2022

We consider a nonlinear stochastic heat equation in spatial dimension $d = 2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $ε > 0$ but divided by a factor of $\sqrt{\log ε^{ −1}}$. We impose a condition on the Lipschitz constant of the nonlinearity so that the problem is in the “weak noise” regime. We show that, as $ε ↓ 0$, the one-point distribution of the solution converges, with the limit characterized in terms of the solution to a forward-backward stochastic differential equation (FBSDE). We also characterize the limiting multipoint statistics of the solution, when the points are chosen on appropriate scales, in similar terms. Our approach is new even for the linear case, in which the FBSDE can be solved explicitly and we recover results of Caravenna, Sun, and Zygouras.

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Published In

Annals of Probability

DOI

EISSN

2168-894X

ISSN

0091-1798

Publication Date

May 1, 2022

Volume

50

Issue

3

Start / End Page

1204 / 1253

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 4904 Pure mathematics
  • 0104 Statistics
  • 0101 Pure Mathematics
 

Citation

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Dunlap, A., & Gu, Y. (2022). A forward-backward SDE from the 2D nonlinear stochastic heat equation. Annals of Probability, 50(3), 1204–1253. https://doi.org/10.1214/21-AOP1563
Dunlap, Alexander, and Yu Gu. “A forward-backward SDE from the 2D nonlinear stochastic heat equation.” Annals of Probability 50, no. 3 (May 1, 2022): 1204–53. https://doi.org/10.1214/21-AOP1563.
Dunlap A, Gu Y. A forward-backward SDE from the 2D nonlinear stochastic heat equation. Annals of Probability. 2022 May 1;50(3):1204–53.
Dunlap, Alexander, and Yu Gu. “A forward-backward SDE from the 2D nonlinear stochastic heat equation.” Annals of Probability, vol. 50, no. 3, May 2022, pp. 1204–53. Manual, doi:10.1214/21-AOP1563.
Dunlap A, Gu Y. A forward-backward SDE from the 2D nonlinear stochastic heat equation. Annals of Probability. 2022 May 1;50(3):1204–1253.

Published In

Annals of Probability

DOI

EISSN

2168-894X

ISSN

0091-1798

Publication Date

May 1, 2022

Volume

50

Issue

3

Start / End Page

1204 / 1253

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 4904 Pure mathematics
  • 0104 Statistics
  • 0101 Pure Mathematics